Seven Square Numbers

Stage: 2 Challenge Level:

Mrs Jopling, a teacher at Heversham St Peter's Primary School
sent us this email:

My class have successfully completed the challenge for
Seven Square Numbers. 12 children went on the internet in pairs and
worked together to find a solution. At the end of the session the
children came together for a plenary session and looked at ways of
pooling resources to solve the challenge as no group had completed
it. Claire worked out that the following numbers squared and added
together, then squared gave a whole number:

It sounds like a lot of team work went into this solution, well
done to you all. Duncan, a student at Balgowan Primary School,
Beckenham went about this problem very logically:

First, to work it out, I put the numbers 1,4,9,16,25
and found all the combinations of 3 of those numbers:
14,21,30,26,35,42,29,38,45,50.
I chose a square number 961 and took away all the combinations to
make these: 947,940,931,935,926,919,932,923,916,911.
Then I wrote down all the squares where their square roots were
between 10 and 20:
100,121,144,169,196,225,256,289,324,361,400.
Afterwards I took the numbers in the 300s (324 and 361) and took
them away from the earlier numbers to leave:
611,570,586,607,593,586,623,630,602,565,595,558,608,571,599,562,592,555,587,550.
Then I worked out the combinations of two of the numbers in the
200s added together and they were 481,545 and 514.
I then took away these numbers from the other ones but I realized
545 and 514 ended up with numbers too low even for the lowest
square number and taken away from the highest number 611 so I used
only 481.
I took this away from 602 to leave 121 and I worked out the answer
was: 121+225+256+324+25+9+1=961.

In other words, 11² + 15² + 16²
+ 18² + 5² +3² +1² =
31². Roger at Our Lady's J.M.I. in Welwyn Garden City
has come up with a different way although you don't explain how you
did it, Roger:

The squares 4,9 and 25 total 38.
The squares of 12, 13, 15 and 18 total 862 add 38 = 900.
The square root of 900 is 30.

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